Optimal Interval Routing in k-Caterpillars and Maximal Outer Planar Networks

1. Optimum Interval Routing in k-Caterpillars and Maximal Outer Planar Networks Gur Saran Adhar Department of Computer Science University of North Carolina at Wilmington, USA

2. Outline of the talk • Research Context • Message Passing Networks • Explicit vs. Implicit Routing • Interval Routing Scheme • Main Contributions • Optimal Interval Routing in K-Caterpillars Maximal Outer Planar Nets. Open Question, References

3. Message Passing Networks • Co-operating parallel processes share computation by way of message passing • Example: MPI processes interface provides • MPI_Send(); • MPI_Recv(); • Different from the shared memory multiprocessing

4. Routing Schemes • Explicit Routing Routing Tables • Implicit Routing Labeling nodes of • chain, • mesh, • hypercube, • CCC, etc…

5. Compare the following two Labeling Schemes for a chain

6. Observation:1 • First labeling defines a total order on the nodes in the chain • Second labeling does not define a total order • Each node receives a unique label

7. Observation:2 • A chain (one-path) is an alternating sequence of: node (a complete set of size one) followed by an edge (a complete set of size two). • Adjacent edges share exactly one node

8. Observation:3 • A chain represents an intersection relationship between INTERVALS on a real line. • A chain is a special tree and the individual INTERVALS its sub-trees • A route is essentially linking the sub-trees

9. Interval Routing • A type of implicit routing • Introduced by Santoro • SK:1985, The Computer Journal • Work by Van Leeuwan, Fraigniaud • LT:1987, The Computer Journal • FG:1998, Algorithmica • Not optimal in general • PR:1991, The Computer Journal • Present Research • GSA:2003, PCDN 2003

10. Interval Routing Scheme-Main Idea

11. Interval Routing Scheme-Main Idea

12. Recursive Definition: tree • Basis: one node is a tree • Recursive Step: adding a new node by joining to one node in the graph already constructed also results in a tree

13. Recursive Definition: K-tree • Basis: A Complete graph on k nodes is a K-tree • Recursive Step: adding a new node to every node in a complete sub-graph of order k in the graph already constructed also results in a K-tree

14. Example: 4-tree

15. Definition: Caterpillar • A Caterpillar is a tree which results into a path when all the leaves are removed

16. Example: Caterpillar

17. Definition: K-Caterpillar • A K-Caterpillar is a k-tree which results into a k-path (an alternating sequence of k complete sub-graphs followed by (k+1)-complete sub-graphs) when all the k-leaves (nodes with degree k) are removed

18. Example: 2-Caterpillar

21. Definition: Maximal Outer Planar Network (MOP) • A network is outer planar if it can be embedded on a plane so that all nodes lie on the outer face • A outer planar network is maximal outer planar which has maximum number of edges

22. Example: Maximal Outer Planar Network

23. MOP as Intersection Graph of sub-trees of a tree

24. Definition: Median • A node is a median if the average distance from every other node is minimized.

25. Dual of the Example Maximal Outer Planar Network

32. MST of Example MOP rooted at the Median

34. Conclusion • New optimal algorithm for k-caterpillars and maximal outer planar networks.

35. References [SK:1985] Labeling and Implicit Routing in Networks, Nocola Santoro and Ramez Khatib, The Computer Journal, Vol 28, No.1, 1985. [LT:1987] Interval Routing, J. Van Leeuwen and R.B.Tan, The Computer Journal, Vol 30, No.4, 1987. [FG:1998] Interval Routing Schemes, P. Fraigniaud and C. Gavoille, Algorithmica, (1998) 21: 155-182. [PR:1991] Short Note on efficiency of Interval Routing, P. Ruzicka, The Computer Journal, Vol 34, No.5, 1991. {GSA:2003] Gur Saran Adhar, PCDN’2003

36. Thank you